## Research Seminars in the School of Mathematical Sciences

The School of Mathematical Sciences host regular research seminars delivered by internal and external researchers on topics across mathematical sciences. Seminars take place on the Kevin Street campus and are open to all.

## Seminar 12/02/16: Ways of Seeing Julia Sets: Visualizing the forces that shape fractal Julia sets

Ways of Seeing Julia Sets: Visualizing the forces that shape fractal Julia sets

Ted Burke

School of Electrical and Electronic Engineering, DIT

Friday 12 February 2016

1pm, Blue Room, DIT Kevin Street

Abstract:

Early in the 20th century, work by mathematicians such as Pierre Fatou and Gaston Julia on complex dynamics led to the definition of so-called Julia sets. When a rational complex polynomial function is applied iteratively to a complex number, z, it produces a sequence of complex values called the orbit of z. Depending on the particular function and on the value of z, that orbit may or may not remain bounded. For a given function, the Julia set forms the boundary between those regions of the complex plane where the orbits remain bounded and those where they do not.

Intriguingly, even for quite simple iterative complex functions, Julia sets often take on very striking fractal shapes. With the aid of a computer, it is easy to visualize the Julia set of a function but, for most people, understanding why it takes on a fractal shape is difficult. In my own ongoing struggle to gain a more intuitive understanding of fractals, I have written many short computer programs to visualize them in different ways. In this presentation, I will explore some ways of visualizing Julia sets which I found helpful in understanding why they are fractal.

## Seminar 11/12/15: Finite k-nets in projective planes

Finite k-nets in projective planes

Nicola Pace

Dublin Institute of Technology

Friday 11 December 2015

1pm, Blue Room, DIT Kevin Street

Abstract:

This talk deals with k-nets embedded in the projective plane PG(2,K) defined over a field K. They are line configurations in PG(2,K) consisting of k≥3 pairwise disjoint line-sets, called *components*, such that any two lines from distinct families are concurrent with exactly one line from each component.

The case k=3 is particularly interesting because, in some instances, it is possible to realise finite groups. A 3-net is said to *realise* a group (G,·) when the following condition holds. If A,B,C are the components, then there exists a triple of bijective maps from G to (A,B,C), say α∶ G→A, β: G →B, γ: G → C, such that a·b =c if and only if α(a),β(b),γ(c) are three collinear points, for any a,b,c∈G. If K has zero characteristic, 3-nets realising a finite group are classified. If the characteristic of the field is p>|G|, then the same classification holds true apart from three possible exceptions: A_{4},S_{4} and A_{5}.

Key ideas and results needed for the classification are presented.

## Seminar 06/11/15: An Examination of the Neural Unreliability Thesis of Autism

An Examination of the Neural Unreliability Thesis of Autism

John Butler

Dublin Institute of Technology

Friday 6 November 2015

1pm, Blue Room, DIT Kevin Street

Abstract:

Reports have recently emerged pointing to the possibility that evoked sensory-neural responses might show greater trial-to-trial variability in individuals on the autism spectrum disorder (ASD). The general notion is that signal averaging procedures typically used in neurophysiological and neuroimaging studies obscure the fact that there are ongoing and presumably relatively dramatic fluctuations in responsiveness to individual events.

Here, we examined this thesis in a matched cohort of typically developing children and children with ASD (N=20), using high-density electrical recordings of visual and somatosensory evoked responses. If the thesis is correct that people with ASD have an unreliable evoked response, a number of straightforward predictions can be made; 1) the averaged evoked response should be broader and have a delayed peak for all components; 2) ASD individuals should have a greater variability of phase dispersion across single trials.

Furthermore, we simulated an unreliable evoked response by introducing temporal and amplitude variability at a single trial level. This simulated data was then compared with the observed TD and ASD data, and illustrated the predictions of the unreliable evoked response and the sensitivity of the measures to small perturbations.

Our data show highly similar reliability in the neural responses to visual and somatosensory stimuli in our matched groups, while the simulated data show differences predicted by the unreliability thesis. These results against a straightforward unreliability hypothesis and instead favours an argument taking into account subtleties and specializations that are present in a complex disorder such as autism.

## Seminar 20/11/15: Mathematical Models of Visual Processing: Explaining Illusory Motion

Mathematical Models of Visual Processing: Explaining Illusory Motion

Garrett Greene

Dublin Institute of Technology

Friday 20 November 2015

1pm, Blue Room, DIT Kevin Street

Abstract:

I will give a brief overview of mathematical models of visual processing in the brain, both in the retina and in the visual cortex. I will then discuss some recent results demonstrating how certain motion illusions can arise from models of non-linear processing in the retina.

The Ouchi Illusion and the Out-of-focus Illusion are stationary images which can produce a perception of motion or "jittering" under certain conditions. These effects are related to small unconscious eye movements, which are ubiquitous in humans and play an important role in human vision. Since we do not normally perceive motion in the world during these eye movements, there must be a mechanism to compensate for the motion of the visual scene across the retina. I will present a computational model of visual processing which compensates for eye-motion under normal viewing conditions. In this model, the effect of the Ouchi and Out-of-focus illusions are predicted as failure modes of the compensation mechanism.

## Seminar 30/04/15: Quantum Fields and Solid State Physics

Quantum Fields and Solid State Physics

Marianne Leitner

Dublin Institute for Advanced Studies

Thursday 30 April 2015

1pm, Room KA3-011, DIT Kevin Street